AD-A015 899
SOLID PROPELLANT KINETICS. V. FUEL-OXIDIZER
REACTION RATES FROM HETEROGENEOUS OPPOSED FLOW
DIFFUSION FLAME
C. M. Ablow, et al
Stanford Research Institute
Prepared for:
Office of Naval Research
Defense Contract Administration Services Region
December 1974
DISTRIBUTED BY:
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U. S. DEPARTMENT OF COMMERCE
ADA015899
296108
December
1974
SOLID PROPELLANT KINETICS
V. FUEL-OXIDIZER REACTION RATES FROM
HETEROGENEOUS OPPOSED FLOW DIFFUSION FLAME*' t
C. M. Ablow and H. Wise
Stanford Research Institute
Menlo Park, California 94025
Reproduction in whole or in part is permitted for
United States Government.
D D C
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^ OCT 16 1975
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This work was sponsored by the Office of Naval Research, Power Branch,
Washington, D.C. , under Contract N00014-70-C-0155.
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4. t1TLE (and Subti(la)
SOLID PROPELLANT KINETICS V. FUEL-OXIDIZER
REACTION RATES FROM HETEROGENEOUS OPPOSED FLOW
DIFFUSION FLAME
7. AUTNOR(j)
C. M. Ablow and H. Wise
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Stanford Research Institute
Menlo Park, California 94025
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19- KEY WORDS (Continue on reverse side if necessary and identify by block number)
Diffusion Flame
Opposed Flow
Propellant Kinetics
Ammonium Perchlorate
Flame Model
20. ABSTRACT (Commut on reverse side if necessary and identify by block number)
A theoretical model is presented relating the gas dynamics and chemical kinetics
of the opposed flow diffusion flame formed in the stagnation region between two
opposing streams of gaseous reactants, one originating from the surface of a sub-
liming solid, such as ammonium perchlorate. At low gas flows the regtession rate
of the solid is controlled by the physical properties of the system, including the
net heat of gasification, the heat of combustion, and the transport parameters.
At high gas flows a limiting solid regression rate is attained due to reaction-rate
>,STn1473
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19. KEY WORDS (Continued)
20 ABSTRACT (Contmuad)
limitations that cause incomplete combustion of the reactants. The theoretical
model developed for the heterogeneous opposed flow diffusion flame allows inter-
pretation of the limit in solid regression rates in terms of global reaction
kinetics. Calculations have been carried out for a range of parameters, including
net heats of gasification and activation energies. For the AP-propylene system
the experimental data can be fitted to a second-order gas-phase reaction rate with
an activation energy of 37 ± 1 kcal/mole and a preexponential coefficient of
t”? -i -l
10^ cc*mol ^sec
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ABSTRACT
A theoretical model is presented relating the gas dynamics and
chemical kinetics of the opposed flow diffusion flame formed in the
stagnation region between two opposing streams of gaseous reactants,
one originating from the surface of a subliming solid, such as ammonium
perchlorate. At low gas flows the regression rate of the solid is
controlled by the physical properties of the system, including the net
heat of gasification, the heat of combustion, and the transport param-
eters. At high gas flows s limiting solid regression rate is attained
due to reaction-rate limitations that cause incomplete combustion of
the reactants. The theoretical model developed for the heterogeneous
opposed flow diffusion flame allows interpretation of the limit in
solid regression rates in terms of global reaction kinetics. Calcula-
tions have been carried out for a range of parameters, including net
heats of gasification and activation energies. For the AP-propylene
system the experimental data can be fitted to a second-order gas-phase
reaction rate with an activation energy of 37 ± 1 kcal/mole and a
13 -1 -1
preexponential coefficient of 10 cc*mol sec
I
Introduction
An understanding of the solid propellant combustion process is
essential for interpretation of motor performance and the prediction of
propellant characteristics under various physical conditions. To
elucidate the burning mechanism of AP-based composite solid propellant >
a number of analyses have been carried out that differ mainly in the
degree of complexity of the theoretical models employed. Common to
all thece models is the mechanism of (1) conductive heat transport to
the propellant surface from the adjoining high-temperature, gaseous
reaction zone, (2) gasification of the condensed-phase component (fuel
and oxidizer), (3) diffuslonal mixing of the gaseous constituents, and
(4) exothermic chemical reaction in the gas phase. Near the solid/gas
interface an AP (ammonium perchlorate) decomposition flame may form part
of the reaction zone involving fuel and oxidizer species emanating from
the solid propellant in their original chemical composition or modified
by intervening chemical process. In addition, subsurface reactions and
phase changes of the solid at the surface may make contributions to the
regression rate of the solid propellant.
The complex nature of the soli propellant reacting system and the
high temperatures involved make the task of elucidating the numerous kin-
etic steps most formidable. However as a first step in the study of the
reaction mechanism it may be adequate to have available information on
the "global" reaction kinetics that offer a measure of the heat release
rate due to exothermic chemical reaction in the gas phase as a function of
temperature, pressure, and gas composition. The availability of such data
should permit semiquantitative interpretation of the reaction rates in
the flame zone, and provide a better measure of the interplay between
diffusional mixing and chemical reaction under various operational conditions.
* 1
References are listed after the appendix.
The opposed-jet experimental technique*" has been used to obtain
kinetics parameters In the homogeneous case where jets of gaseous fuel
3
and oxidizer support a diffusion-controlled flame. As the velocity of
the jets is increased, a smaller fraction of the incoming reactants are
consumed due to kinetic limitations. The unburnc reactants act as diluents
carrying away heat from the flame so that at a high enough velocity the
flame is extinguished.
The same technique has been applied to the heterogeneous case where
•1
a jet of gaseous fuel is directed against a subliming solid oxidizer.
As the velocity of the jet is increased, the solid regression rate has
been observed to increase to a certain limit, at which a further increase
in jet velocity leaves the regression rite unchanged. This limit has
been identified to correspond to the maximum fuel consumption rate
for the system. At greater flow rates, fuel slips through the flame
zone to act as diluent and heat sink near the subliming surface, thus
limiting the surface temperature and regression rate. A theoretical model
is developed that relates stagnation flow and regression rates to global
kinetic parameters and the heat requirement at the subliming surface.
In the following sections we have prepared first a descriptive
presentation of the model, tne approximations made, and the results
obtained. This part is followed by Appendix A containing the details of
the mathematical analysis and Appendix B giving the application 10 the
AP-propyiene system.
2
II Theoretical Model
The objective of the theoretical analysis is to relate the rates of
chemical reaction occurring in the gas-phase diffusion flame to the
combustion characteristics cf the heterogeneous opposed-flow diffusion
flame (HOFD) set up between a subliming solid oxidizer and a gaseous
jet of fuel with inert diluent. Alternatively, the analysis allows
deduction of reaction-rate parameters of the gas-phase deflagration
from experimental measurements of the HOFD.
The cylindrically symmetric, steady flow field is sketched in
Figure 1. It has been assumed that the ratios of jet nozzle radius and
distance from the solid to solid stick radius are so large as to be
effectively infinite. Although there are two coordinates, radial distance
r from the axis of symmetry and axial distance z from the surface of the
solid, the flow along the axis is decoupled from the remainder of the
flow if one assumes that the radial diffusion of mass and heat is negli-
gible in comparison with axial diffusion and convection. The distributions
of temperature and species along the axis are then determined by one-
dimensional equations in z only.
The multi-step kinetics of the gas-phase deflagration process are
approximated by a global reaction-rate expression. Such an approximation
appears to be an adequate quantitative description of the kinetics in
terms of local heat-release rates. Obviously it is less satisfactory in
terms of an analysis of spatial distribution of chemical intermediates
since no detailed account is taken of the mechanism of the reaction and
the intermediate species produced.
In Appendix A the conservation equations for mass and energy are
formulated ir. terms of the Shvab-Zeldovitch model with constant, temperature-
averaged parameters for the physical and transport parameters, and a Lewis
3
number of unity. Such a model is based on a single overall reaction
involving two gaseous reactants. The coefficients in the equations
depend on the axial component of the mass-flux distribution, which in
turn is determined by the momentum balance equations. However in the
present system the momentum variations are expected to be small so that
almost any flow satisfying the mass conservation equation should be an
5
adequate approximation. Spalding and others have assumed, for the case
of opposed gaseous jets, a linear variation of the radial velocity com-
ponent with distance from the axis and a similar variation of the axial
component irom the plane through the stagnation point and perpendicular
to the axis. This flow can be strictly correct only near the stagnation
7
point. Seif-similar boundary layer flow has also been employed. This
flow has the advantage of providing a solution of the complete set of
flow equations. However, it seems inapplicable to the present case where
a large regression rate disrupts the boundary layer.
In the present model the axial component of the mass flux is described
by a series approximation with a sufficient number of terms to satisfy
the conditions at the solid surface and at the inlet for the gaseous
reactant. The flow has a stagnation point and exhibits the correct
behavior in its neighborhood. The radial component of the flow is deter-
mined by the axial mass flux gradient and has in the model the constant
sign that implies a realistic radial outflow everywhere. To provide an
analytical model for rapid computation and evaluation of different
parameters we approximate the temperature distribution by a polynomial
in the distance variable. An advantage of the analytic approximation
8
method over asymptotic methods is that no parameter need take extreme
values for the approximation to be useful.
In +he analysis of HOFD information is needed on the conditions at
the gas/solid interface. For the present model, we have employed exper-
10
imental results relating surface temperature and regression rate.
4
Ill Analytical Results
The most striking feature of the experimental results for the AP/fuel
4
system is the observed change from a strong variation of solid regression
rate at values of the fuel mass flux to nearly constant regression
rate at higher fluxes. The theoretical model interprets this behavior
to be due to a change from complete fuel consumption in one case to
unburnt fuel slippage through the flame to the solid surface in the other.
In the latter case the unburnt fuel acts as diluent and heat sink.
At very low fuel flow rates the thin-flame approximation applies
(Appendix A. 9). In this regime the combustion process is controlled by
the mass transfer rate of reactants to the flame and is a sensitive
function of the heat L of gasification of the solid reactant. The
value of L is given by the difference between the heat of sublimation
and the exothermic heat of reaction at the solid surface. A series of
calculations have been carried out over a range of values of L. The
sensitivity of the system to variations in L is exhibited by the data
presented in Figures 2 and 3. The lines going through the origin depict
the variation of m as a function of m at different weight fractions
S G
of fuel (Y ) for two values of L. In applying those data to the expert-
1X5 4
mental results obtained for the AP/propylene system one deduces a value
of L = 87 ± 1 cal/gm AP (Table 1). It is to be noted that this value
is considerably less than the heat of sublimation of AP (450 cal/gm AP),
an indication of the contribution of exothermic reactions in the condensed
phase. Similar conclusions were drawn in previous studies of the tempera-
11
ture distribution in solid AP during steady-state deflagration and in
12
a modelling analysis of AP combustion.
The kinetic-controlled regime of the HOFD system is demonstrated by
the attainment of the solid regression rate limit at high mass fluxes of
5
rpsz 33Ctti55p&3
gaseous fuel. This condition is represented by the intersection of the
straight lines with the curves in the (m , m ) plane of Figures 2 and 3
G S
computed for different values of L and the activation energy E for
a single-step reaction of first order in fuel and oxidizer. These data
indicate that with increasing E the transition to kinetic control occurs
*
at progressively lower fuel mass fluxes (critical value = m \ However,
G
an increase in the initial fuel concentration (Y ) exhibits a somewhat
FG
more complex pattern. For example the data in Figure 1 indicate that
$
for E = 25 an Increase in Y from 0.25 to 1.00 causes m to decrease
* FG G
progressively while m appears to go through a maximum as observed
4 S
experimentally. This behavior may be due to the progressive departure
from stoichiometric conditions (on the fuel-rich side) as Y increases
FG
so that the flame cools and the regression rate decreases.
While the computations for Figures 2 and 3 consider all the oxygen
in the AP to be available for C H -oxidation in the flame (Y = 0.547),
3 6 SX
those of Figure 4 consider some of the oxygen consumed for ammonia
oxidation (Y = 0.400). These computations point to the marked effect
bA
of the oxidizer weight fraction at the solid surface on the combustion
process. One also notes in Figure 4 that the reduction in the supply
of oxidizer has moved the maximum regression rate toward lower values
of Y , as expected on the basis of a departure from stoichiometry.
FG
The theoretical model is applied to the experimental data* for the
AP/propylene system. As can be seen the two-regime model represents
a satisfactory approximation to the experimental observations (Figure 5).
*
The limiting mass flux m for each weight fraction of fuel has been
S
recorded in Table 1. This data is used to compute the activation energy
E (Appendix A. 10). The value for E so obtained should be independent
of the fuel weight fraction, as is found to be the case (Table 1). The
experimental results yield an activation energy of 37 x 1 kcalAiiole for
6
i
i
the gas-phase reaction associated with the combustion process of AP
*
and propylene. Thus the rate constant for this reaction may be written
13 -1 -1
as k = 10 exp(-37000/RT) ccmol sec
IV Conclusions
The opposed flow diffusion flame is a convenient and precise tool for
the investigation of kinetic parameters at high temperatures. In the
homogeneous case of opposed gaseous jets, the flame is abruptly extinguished
on the axis of symmetry when the reactant flow reaches a critical value;
extinction occurs when the reactants are no longer completely consumed and
therefore act as diluents carrying heat away from the flame. A model of
3
the flow and reaction has been used to derive overall reaction parameters
from the opposed flow data.
In the heterogeneous case one stream is produced by gasification of
a condensed phase reactant, the other originates as a gas. A model for
the HOFD at very low flow rates has been used to relate the ratio of
solid and gas mass fluxes to the heat of gasification. Its constancy as
the gas composition is varied indicates that a useful explanation of
the relation between regression rate and heat of gasification has been
found .
Our model further suggests that the levelling-off in observed regres-
sion rate curve as the gaseous reactant flow is increased is due to in-
complete combustion. These critical conditions can be related to the
gas phase reaction rate. The model predicts activation energies that
are independent of the gas stream composition. Thus the HOFD offers a
unique approach to the evaluation of kinetic parameters of concern to
complex combustion systems such as those encountered in solid propellant
deflagration.
*
For Y = 0.400 or.e finds a value of E = 24.5 - 2 kcal/molc AP.
XS
7
Appendix A
EQUATIONS OF THE MODEL
A.l Mass and Energy Conservation Equations
For the one-dimensional model employed in our analysis we use the
13
Shvab-Zeldovitch equations which govern mass and energy conservation
in convective, diffusive, reactive, steady-state flow. To reduce the
degree of complexity of the problem without materially affecting the
validity of the model the simplifying assumptions are made that the
binary diffusion coefficient D is the same for ail species and the Lewis
number is unity, so that heat conductivity X. is related to the gas density
p and average specific heat C by - pD C , We may write therefore:
P P
7. (ov Y - pD 7 Y ) = w
i i l
V. (pv C T - pD C 7 T) = -F. h° w
P P i i '
where v is the velocity vector, w ^ is the rate of formation of species i
in the single step reaction, Y^ is the mass fraction and h^0 the heat of
formation of the species, and T is the temperature. If C is the
pi
specific heat at constant pressure of species i then C is defined by
P
C T = F Y |’T C dT ,
p i i Jo pi
The stoichiometry of the reaction may be expressed as :
F v/ m - E v" m ,
1 i i i
where lvi^ is the chemical symbol for species i and the and are
stoichiometric ratios. Then
w = W (v - v ) U) ,
i i i i
8
where is the molecular weight of species i and molar reaction rate
is independent of i. The mole fraction of species i is :
= (Y,/W ')/Z (Y /W, ) .
i i i k k
Define the heat of reaction Q by
Q = -Z h° W (v" - v7 )/w ,
i i i i o
where W is a constant molecular weight. Then the conservation equations
o
read
L (« ) = L (a? ) = a) W ,
i To
where
or = Y W /W (v" - v7 )
i i o i i i
« = C T/Q ,
T p
and differential operator L is defined by
L(o) = pv.V a - v.p Etftf •
The equation for conservation of total mass ,
V.(p?) = 0 ,
has been used to simplify L .
A. 2 Axis Approx imatio.i
On the axis of cylindrical symmetry vector v has just the one
component v in the z-direction so that
da _
L(a) = pv t— - T'PD?01' .
oz
9
Neglecting radial diffusion gives the one-dimensional equation
L(o0
d da;
pD
dz dz
The equation has a simpler form if the axial coordinate is changed
from z to the dimensionless coordinate y :
y = m JZ (1/pD) dz ,
o o
where m is a constant mass flux. Then
da
dy
For definiteness m is taken to be m ,
o G
the mass flux at the gas inlet,
m_ - ~pv at z = ® .
G
and the origin of coordinates, z=0, is put at the surface of the solid.
A. 3 Flow Field
If s stands for any of the differences (a - a ) or (a - a ) then
IT i j
L(s) = 0
or
ms - s = 0, m = pv/m
where the prime denotes differentiation with respect to y . This can be
solved for a non-constant solution function s if m is given as a function
of y . Conversely if s is given, m is determined.
In order that s may replace y as a coordinate, one requires s/ ^ 0.
Also, for convenience,
s = 0 at y = 0 ,
s = 1 at y = 00 ,
so that s' is positive. Conditions on m are
m = m /m at y = 0
S G
m = -1 at y = 10
where m is the mass flux at the surface of the solid. The y coordinate
S
of the stagnation point, y , is found by solving
o
m = 0 at y = y .
o
In the experiment, there is a cylindrical nozzle for the gas inlet
and the oxidizer has the shape of a solid cylinder. The ratios of
solid diameter to nozzle diameter and distance to the nozzle sre
parameters controlling the flow configuration, in general. However,
both ratios were taken sufficiently large in the experiments so that
no effect of further variations was observed. Thus both ratios are
essentially infinite, and the geometrical configuration does not enter
the calculations explicitly.
In order that m = -1 at y =*, function s' should be proportional
to exp(-y) when y is large. A function of this sort is
s = 1 - exp(-y - f)
with f bounded at infinity. A convenient choice is to make f a
polynomial in [y/(y + b)] where b is a constant taken to be positive
so that the flow region is free of singularities. One finds
s' = (i - s> a + f'>
m = f /(I + f ) - (i + f )
m' = f"'/(l + f') - [f"/(l + f’’)]2 - f"
Conditions at infinity have been satisfied. We take f to be quadratic:
f = a + a x + a x ,
o 1 2
x = y/(y + b) .
Differentiation gives
f' = 1(1 - x)2/b] (a^ + ^»2x)
f = [2 (1 - x) /b
I (a - a - 3a x)
2 1 2
f"' = [6 (1 - x)Vb3] (&i - 2a2 + 4a2x)
The conditions at y = 0 give
a = 0 ,
o
a = a + (a + b)(a + Mb)/2 , M = 1 + m /m ,
2 111 S G
s = 1 + a /b ,
o 1
m' = 2 [3 (b + ai)(ai - 2aa> - 2 (u? - a,)'
2 2 2
(b + ai> (a2 - a^)]/b (b + aj) (
where s is the value of s at s = 0 and m is the value of in there,
o S
The requirements that b and s^' be positive restrict the choice of
a and a to the region -b < a *" a . In addition, a physically
12 12
reasonable flow has m negative throughout. Ensuring this in general
seems difficult ; however, if one lakes a2 - ~(l/2)a^ and restricts
9
12
imwijwini'i'mmgff—uM,
a^ to -b < a < 0 , then the formulas simplify and one can readily see
that this condition is satisfied.
The simplified formulas obtained when a = -(l/2)a read
2 X
a x (1 - x/2) ,
1
(ai/b)(l - x) ,
2 4
-3 (a^/b )(1 - x) ,
, 3 5
12 (a^/b )(1 - x)
A. 4 Boundary Conditions
If T is the surface temperature of the solid and T the ambient
S A
temperature, the heat flux into an inert solid in the steady state is
dT
— = rn C <T - T) ,
S dz S S S A
where X is the conductivity and C is the specific heat of the solid,
S S
Equations for the conservation of heat and species across the surface
read
dT
dT
\ — = m L + A. — I
dz S S dz S
dY.
-PD
dz
ms <YSi ~ V *
where L is the net heat of gasification of the solid and Y is the
ol
mass fraction of species i in the vapor given off by the solid.
Define 0/ to conform to Q < ;
Si i
a = Y W /W (V - v ) .
Ci Si o i i i
13
Then the species boundary condition becomes
m or
i
”s <ai - V “ y 0 •
The heat condition for the inert solid reads
°T ’ <VV ta/«> * <CS V ‘“tS - V’ “ J = ° •
where the variation of C with temperature and concentration ratios
P
has been neglected. If there is heat release by reaction in the solid
phase, the heat condition has a different form. Only inert solid is
treated here.
At the gas inlet, the temperature and species concentrations have
known values :
ot = a , or = a , at s = 1 ,
T TG i iG
A. 5 Species Distributions
The distribution of species i is related to that for temperature
by
a = a - A - B s ,
i T i i
where A^ and are constants determined by the boundary conditions.
One finds
A = of -Of
i TS iS
A + B = Of -a
i i TG iG
s ' B = (m /m ) UL/Q) + (C /C ) (or -or ) - -or )] .
o i S G S p TS TA iS Si
These conditions are used to determine A , B , and or for given s . M,
i 1 iS o
and or
TS
14
A. 6 Reaction Rate Equation
The Arrhenius gas phase reaction rate expression may be written as
n -E/R T _ , n .
• u) = BT e o FI (X p/R T) 3 ,
3 o
where E is the activation energy, R the gas constant, B and n are
o
constants, n. is the order of the reaction with respect to reactant j,
0
and index j ranges over the species consumed in the reaction. Here
j = F or X .
Let W be the average molecular weight ,
W = 1/Zfl^/W > .
Then X = Y W/W so that the X can be eliminated. Neglecting the
3 3 _ 3 3
variation in W allows one to set W = W. Since Y and T are proportional
o 3
to the variables of the theory, « and ot one obtains
j T
» . B (p/s «pyc)n-aj . V/<vt n ,
O T p j j
E* = E C /R Q .
P o
The equation for conservation of energy,
L (a^) = uj w ,
simplifies when the independent variable y is replaced by s to read
2 2
O' ** + R = 0 , R = (w W) [pD/m^ (s') ] ,
T G
where the dots indicate differentiation with respect to s.
The dependence of pD on the temperature is given by
PD = o D (T/T )
A A A
15
m
m
Then
_ t i -2 n+d-En-i -E /^m n n;
R = B (s ) or Je Tri(A+Bs-»)J
T 3 3 T ’
b' = (BW/m 2) p Dl'd (p/R >&li (Q/C )n+d_l:nj J] (V ')nj
G AAA o p j
Boundary conditions
a = or
T TC
on 0/ read
T
at s = 1 ,
(m /m ) [(L/Q) + (C /C )
S i S p
(a - <* )] at s = ° .
T TA
The curve in the (s, & ) plane representing the solution lies
inside the polygon defined by
0*3*1, J^or , a A . + B s
T T 3 3
~._tc lunction R is positive inside the polygon and is zero on the sides,
except the sides s = 0 and s = 1. The differential equation shows that
the second derivative or, for brevity, "curvature" of the solution
curve is equal to (-R). Where reaction occurs, rate function R is
positive and the solution curve is convex above, i.e., arched.
rs
f:
16
I
L.
I
§
A. 7 Approximate Temperature Distribution
Possible solution curves for dimensionless temperature O' as a function
of distorted distance coordinate s are sketched in Figure 6. Point A repre-
sents the thin, diffusion-controlled flame for which the reaction rate is
so fast relative to the rates for diffusion and convection that both react-
ants are completely consumed at the flame surface. For this case, reaction
rate function R is infinitely large in the interior of the polygon, but is
zero on its sides.
For finite, decreasing values of R the reaction zone broadens,
and temperature curves are found like those labelled 1 to 4 in
Figure 6, Point V, where R is a maximum, has been taken in the model as
f..e flame location. It should be close to the points of maximum temper-
ature and maximum reaction rate.
The temperature distribution is approximated in the model by a
polynomial in s that fits the energy conservation equation at three
points , the two eeges of the flame zone and a point V at the "center"
of the flame where the reaction rate is a maximum. A polynomial of
fourth degree is required to fit the second order equation and its two
boundary conditions :
2 3 4
= b+bs+bs+bs+bs.
T o 1 2 3 4
At Point V, we have
a = l. b s
TV j V
.1=0
£ j<j - 1) b. syJ 2 + Ry = 0
R + R 4- j b s
,sV .ttjV j V
j-1
17
where, in the last equation, the subscript commas indicate partial
differentiation. From the formula for reaction rate function R one finds
R = (- 2m + n B /Z + n B /Z ) R ,
,s F F F X X X
, 2
R = [E /a + (n + d - n - n )/ot - n /Z - n /Z ] R ,
,<*t T F X T F F X X
Z = A + B s - O' ,Z=A+Bs-Q'.
F FF T X X X T
The remaining equations and boundary conditions needed to determine
the b. depend on which type of solution curve is applicable. If both
J
fuel and oxidizer are completely consumed, the curve will be like Type 1
as sketched in Figure 6 . In the limit where the fuel is just consumed
before reaching the solid surface, the curve is of Type 2. The equations
that follow are written for Type 1 and apply to its limit, Type 2. The
other types, with incompletely consumed fuel, were not used in the
comparisons with the experimental observations.
If point (s , of ) is at the inlet edge of the reaction zone,
X TX
a ' = £ b s 0
TX j X
a = A + B s
TX XXX
^ . . J-l
£ J b0 sx ■ Bx
£ j (j - 1) b S
J A
3-2
= 0
where the model polynomial passes through the point according to the
first equation, the point lies on Y =0 according to the second equation,
X
in the third equation the curve is given the slope of Y = 0 since slope
is proportional to heat flux and is therefore continuous, and in the
18
fourth equation the curve is given zero second derivative as determined
by the differential equation being fitted. At the other edge of the
reaction zone the same equations are valid if subscript X is replaced
with F.
The 11 equations above may be solved for the 11 unknowns: the
5 b 's and the pairs of coordinates of Points V, F, and X.
j
A. 8 The Solid-Gas Interface
In an experiment with a known reactive system the material, kinetic,
and configuration parameters would be known. If conditions permitted a
steady-state flame, values for the surface temperature, flame location,
and regression rate could oe measured. A complete model therefore
requires that these quantities be computable. It has been shown above
that the regression rate is determined in the model if the surface
/
temperature and parameter are given.
Since the details of the gasification process at the solid surface
are not iuiown in quantitative detail, the model has been completed by
10
using the empirical relation between surface temperature T,(°K) and
S
regression rate:
2
m, = 1.8 exp (14.9 - 16500/T ) (gm/cm -sec).
b S
Specification of s' has been achieved at two limiting points by use
o
of the extra conditions valid there. The thin-flame limit is discussed
in the next section. The other limit is where the reaction zone just
reaches the surface of the solid. The extra condition, s = 0, is an
F
indirect specification of the free parameter. This limit gives a change
of flame character in the model and is taken to coincide with the ob-
served changes in rate of change of regression rate with inlet flow shown
in Figure 5.
19
A. 9 Low-Flow Regime
If the inlet gas mass flux m is small, a steady flow solution has
G
very low reaction rate almost everywhere since the reaction rate
t -2
function R is proportional to cu(m s ) and R and s are bounded. This
G
condition Ls the thin-flame approximation without any fuel on one side
of the flame or any oxidizer on the other. In particular, at the surface
of the solid, the slope of the solution curve ot' is related to the
TS
other parameters and s by
/ /
3 = Of /B .
o TS F
The pressure variation due to flow acceleration is
3 - /
-p = pv v = m (R Q/p DC) m(npf ) .
z zGoAAp T
Thus p goes to zero with m due to dimensional considerations. It
z G
seems reasonable that the nondimens ional flow pattern also contributes
to the smoothing out of the pressure variations. It is therefore
assumed that (rt* ) also goes to zero with m at the surface of the
T G
solid . This gives
/ t
m O' + on /m ) O' = 0 .
S TS S G TS
/ /
The flow model provides an expression for m^ in terms of s and
(m /m ) :
S G
m' = - (s + M-l ) [3M - 3 + (M + 5) s' - 2(s' )]/3(l - s' )
So o o
o
where M = (m /m ) + 1 . One may then solve for the value of (m /m ) in
S G S G
the low-flow limit:
m /m = 3 [3 - or (l + 0)(l + 20)1/3 [33 + a (1 + 3 ) (1 - 2&)1
S G
where
ci = ci / b
TS F
B = ci -a -a
F TG FG TS
3 = [L/Q + (C /C ){a - * )]/Br .
S p TS TA F
and, if Tc = Ta , »TS = UTA • If « « 1 the relation simplifies to
s = ci ■ f v (9 + <1 m /m )/3 .
S G
A. 10 Calculation Method
For a specified reaction system the calculation requires knowledge
of the species present, the reaction stoichiometry, the ambient and inlet
temperatures, the weight fraction Y of fuel in the inlet stream and the
FG
weight fraction Y of oxidizer in the solid. Values for the specific
SX
heats, C and C , and the heat of combustion Q are also known so that
s p
the- nondimensional quantities ct a ./ are determined. In the low
TA TG FG
flow limit (Section A. 9), if T = T , one has = <v so that B is
A C- TS TA F
known and the relation between heat of gasification I. and low-flow mass-
flux ratio m /m is fixed. The observed value of this ratio determines L.
S G
For the case of complete combustion of the gaseous fuel and no
fuel in the solid, one has Y = Y =0. The mass-flux ratio m /m
FS SF S G
is assumed to remain constant up to the limit of complete combustion.
10
For each m , the empirical regression relation determines the surface
S
/
temperature T . The values of A^, and B^, and s are then computable
21
from the boundary conditions for the fuel species at the solid surface
given in Section A. 5. The three oxidizer boundary conditions then serve
to compute A , B , and ot or Y . the mass fraction of oxidizer in the
X X XS XS
gas at the solid surface.
For specified kinetic parameters, the reaction rate function R is
now a computable function of s and or . The conditions on the temperature
distribution parameters given in Section A. 7 can be readily reduced to
a pair of nonlinear equations for s and s , the s coordinates of the
F X
edges of the reaction zone. (At s the fuel concentration drops to zero;
F
while at s^ the oxidizer concentration drops to zero.) The equations have
been solved for s and s by a two- variable version of Newton's method.
F X
The computations were carried out for different values of the
regression rate m . As the limiting solution that value of m was
S S
selected for which the reaction zone extended to the solid surface so
that s =0,
F
i
I
t
i
|
22
Appendix B
APPLICATION OF THEORETICAL ANALYSIS TO THE AP-FUEL SYSTEM
To demonstrate the suitability of the theoretical model for evalu-
ation of reaction kinetics of a heterogeneous combustion system we have
carried out a series of calculations for the opposed-flow diffusion flame
of solid AP-gaseous propylene, for which experimental data are available.
The reaction is considered to involve the thermal decomposition of AP
during the gasification process with subsequent combustion between the
oxygen produced and the propylene added in accordance with the
stoichiometry
C H + 1/2 0 3 CO + 3 H 0
3 6 2 2 2
In the current analysis the contribution of fuel from the subliming solid
(AP) is taken to be small compared to that introduced from the gaseous
fuel side (CH ). Consequently we do not consider explicitly the forma-
3 6
tion of a premixed flame in close proximity of the solid surface due to
the reaction of the decomposition products of AP (NH and HC10 , or its
3 6
oxidizer intermediates) as postulated in the granular diffusion flame
14
model. Such exothermic reactions are buried in the gasification term
used in our analysis and together with solid-pnase exothermic reactions
contribute to a reduction in the absolute value of the heat of sublima-
15
tion of AP from 480 cal/g AP to 87 cal/g AP. However, in order to
examine the effect of oxygen concentration Y at the solid surface
bX
on the reaction kinetics of the diffusion flame we have carried out
several computer calculations at two levels of Y , one for Y = 0.547
SX SX
corresponding to all the oxygen in solid AP, the other at Y =0.4
SX
corresponding to some oxygen depletion (due to reaction with ammonia).
For the conditions prevailing at the gaseous fuel side we have selected
three fuel weight fractions, i.e., 0.32, 0.60, and 1.00 corresponding
to 24, 50, and 100 vol% of propylene, as employed during the experimental
study. The remaining input parameters for the computer calculations are
listed in Table 2.
A comparison of the present two-regime model with the experimental
4
data is made in Figure 5. The model is fitted to the data at the two
extremes of m near zero and m at its limiting value in .
REFERENCES
1. Reviews are to be found in J. A. Steinz, P. L. Stang, and
M. Summerfield, AIM Publication 65-658, and J. S. Ebenezer,
R. B. Cole, and R. I. McAlevy, III, Technical Report ME-RT 73004,
Steven', Inst. Technology, Hoboken, N. J., June, 1973.
2. A. E. Potter and J. N. Butler, Amer. Rocket Soc. J., 29, 54 (1959).
3. C. M. Ablow and H. Wise, Combustion and Flame, 22, 23 (1974).
r 4. S. J. Wiersma and H. Wise, "Solid Propellant Kinetics. IV.
Measurement of Kinetic Parameters in Opposed Flow Solid Propellant
Diffusion Flames," Interim Technical Report, Contract N00014-70-C-0155 ,
Office of Naval Research, Power Branch (December 1973),
5. D. B. Spalding, Amer. Rocket Soc. J., 31, 763 (1961).
6. F. E. Fendell, J. Fluid Mech. , 21, 281 (1965).
7. L. Krishnamurthy and F. A. Williams, "A Flame Sheet in the Stagnation-
Point Boundary Layer of A Condensed Fuel," paper presented to Western
States Section, The Combustion Institute, Tempe, Arizona, April 1973.
8. A. Linan, "Asymptotic Structure of Counterflow Diffusion Flames for
»» '
Large Activation Energies, Instituto Nacional de Tecnica Aeroespecial ,
Madrid (1973) unpublished.
9. C. Guirao and F. A. Williams, "Models for the Sublimation of Ammonium
Perchlorate", Paper 69-22, Western States Section, The Combustion
Institute, China Lake, April 1969,
1C. J. F. Lieberherr, 12th Symposium (International) on Combustion
(1968), 533-541.
11. H. Wise, S. H. Inami, and L. McCulley, Combustion and Flame 11 ,
483 (1967).
12. C. Guirao and F. A. Williams, "A Model for Ammonium Perchlorate
Deflagration between 20 and 100 atm," AIAA J. Vol. 9, pp. 1345-1356
(1971).
13. F. A. Williams, Combustion Theory, (Addison-Wesley , Palo Alto, 1965).
14. M. Summerfield, G. S. Sutherland, M. J. Webb, H. J. Taback, C. P. Hall,
"Burning Mechanism of Ammonium Perchlorate Propellants", in Solid
Propellant Rocket Research, M. Summerfield, ed.,(Acad. Press, N.Y. 1960).
15. S. H. Inami, W. A. Rosser, and H. Wise, Combustion and Flame, 12,
41 (1968).
2b
Table 2
INPUT PARAMETERS
Quantity
Symbol
Value
Molecular weight-fuel
WF
42 gm/mol
-oxidizer
wx
32 gm/mol
Pressure (1 atm)
P
0.0242 cal/cc
Gas conductivity at
ambient conditions
X
A
—5
6.0 x 10 cal/cm-sec- °K
-3
Gas density at ambient
conditions
PA
1.25 x 10 gm/cc
Lewis number
Ypa da 5p
1
Ambient temperature
ta
300 °K
Inlet temperature
tg
300 °K
Solid density
PS
1*8 gm/cc
Temperature dependence
of pD
d
0
of B
n
0
Order of reaction
for fuel
n
T?
1
for oxidizer
r
nx
1
Stoichiometric coefficient
for fuel
v'
F
1
for oxidizer
V7
X
4.5
Specific heat - solid
"s
0.25 cal/°K
- gas
n
0.25
Arrhenius preexponential
p
B
]0 —1 —1
10AO cc»mol sec
27
SOLID
OXIDIZER
l
i
e
|
II II
T A-8378-48
FIGURE 1 SCHEMATIC CROSS-SECTION OF HOFD FLAME
£
28
10* (gm/cm
\t FIGURE 2 THEORETICAL REGRESSION RATE ms VERSUS INLET FLOW mG LINES AND
| COMPLETE COMBUSTION LIMIT CURVES (Yx$ = 0.547, L = 87 cal/gmAP)
29
10
2 2
mG *10 (gm/cm -*»c)
TA-8378-47
FIGURE 3 l.'EORETICAL REGRESSION RATE m$ VERSUS INLET FLOW mG LINES AND
COMPLETE COMBUSTION LIMIT CURVES (Yxs = 0.547. L = 85 cal/gmAP)
30
I
0 | 1 *
TA-8378-44
FIGURE 6 NONDIMENSIONAL TEMPERATURE aT AS A FUNCTION
OF DISTORTED DISTANCE s
33
32
0
1
%
TA-S378-44
FIGURE 6 NONDIMENSIONAl TEMPERATURE oT AS A FUNCTION
OF DISTORTED DISTANCE s
33
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